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In this paper, we mainly employed the idea of the previous paper to study thesharp uniform $W^{1,p}$ estimates with $1<p\leq \infty$ for more generalelliptic systems with the Neumann boundary condition on a bounded $C^{1,\eta}$domain, arising in homogenization theory. Based on the skills developed by Z.Shen and by T. Suslina for different purposes, we also established the $L^2$convergence rates on a bounded $C^{1,1}$ domain and a Lipschitz domain,respectively. Here we found a "rough" version of the first order correctors(see Theorem 1.3), It allows us to skip the corresponding convergence resultson $\mathbb{R}^d$ that are the preconditions in T. Suslina's papers.

Uniform Regularity Estimates in Homogenization Theory of Elliptic Systems with Lower Order Terms on the Neumann Boundary Problem